Home Page of Aaron Naber

Position

Information:

Office:
Lunt 220

email:
anaber(at)math.northwestern.edu

Areas of
current research:My current research
interests focus on the study of geometrically motivated
equations and their applications. This includes
work in the areas of Ricci curvature, nonlinear harmonic
maps, yang-mills, minimal varifolds, sectional curvature,
ricci solitons, mean curvature flow, Ricci flow, and general
elliptic equations. A rough outline of some of my
research is below, see CV for a more complete and updated
list.

Yang-Mills and Energy Quantization:

Energy Identity and L^1 Hessian
Conjectures: If
A_i are a sequence of stationary Yang-Mills connections,
then one can pass to a weak limit A_i->A. In
such a limiting process one inevitably must deal with
blow up, and a wonderful way to understand this is
through the defect measure. That is, if F_i are
the curvatures of A_i, then one can show that the
measures |F_i|^2 dv_i -> |F_A|^2dv + e(x)dH^{n-4}_S,
where S is an n-4 rectifiable set and H^{n-4}_S is the
hausdorff measure on S. In short, the sequence A_i
may only blow up along a well behaved n-4 dimensional
subset, and when this happens the energy concentrates
along this subset. The energy density e(x)
measures how much energy was lost in this blow up, and
it has been a conjectural picture (which has been proved
in many special cases) that e(x) may be computed
explicitly as the sum of bubble energies arising from
the sequence at x. In [NV16]
together with Daniele Valtorta we prove this to hold in
generality. Additionally, we show that for a
general stationary connection A there is an apriori L^1
hessian bound \int |\nabla^2 F_A|^2 < C on the
curvature. These two points are actually proved
simultaneously. The proof involves a new
quantitative bubbletreedecomposition, which decomposes a general
fixed solution into so called quantitative annular and
bubble regions in an effective manner. The most
difficult analysis is on the annular regions, and
dealing with them involves introducing a new type of
gauge condition, which we call an eps-gauge. This
eps-gauge generalizes the coulomb gauge, but exists even
over singular regions, and allows us to do analysis over
the singularities.

Stratification and Regularity Theory:

Rectifiability of Singular Sets of Nonlinear Harmonic Maps:
If f:M->N is a stationary harmonic map, then one can define
the stratification S^k(f) = {x: no tangent map at x is k+1
symmetric}. It is a classical result of Schoen/Uhlenbeck
that dimS^k <= k, however essentially nothing else has been
understood in generality. If f is also assumed to be
minimizing and N is an analytic manifold, then it has been
proven by Simon that S^{n-3} is rectifiable. Together with
Daniele Valtorta, in [NV15]
we prove for a general stationary harmonic map into any
target that all the S^k are k-rectifiable, which is roughly to
say that S^k are k-manifolds away from a set of measure
zero. The techniques of this paper include the
introduction of a new W^{1,p}-Reifenberg and
rectifiable-Reifenberg theorem, which are of some independent
interest. Roughly, these give criteria which ensure that a
set is rectifiable with measure estimates. In order to
apply these theorems to the singular set, we prove a new
L^2-subspace approximation theorem for stationary harmonic
maps. It is important in the argument that we do not
deal directly with the stratification, but instead provide
effective estimates for the quantitative stratifications, a
notion first introduced in [ChN]. It is
worth pointing out that the techniques are very general, and as
a vague rule tend to work whenever a Federer dimension reduction
argument works.

n-3 Finiteness and Sharp Schauder Estimates for Nonlinear
Harmonic Maps: If f is further assumed to be a
minimizing harmonic map, then in [NV15] we get
effective estimates on the n-3 Hausdorff measure of the singular
set. More analytically, we prove |\nabla f| has effective
estimates in L^3_{weak}, which is sharp since there are examples
which do not live in L^3. The results are related to the
rectifiability methods above, as we in fact prove for even
stationary harmonic maps very effective estimates on the
quantitative stratifications. These estimates hold for any
stationary map, but in the case of minimizers one can show for
the top stratum that the quantitative stratification and the
classical stratification agree, and hence one obtains effective
estimates on the singular set itself.

Structure of Singular Sets of Varifolds with Bounded Mean
Curvature: As another example of the above
technique, if I^m is an integral varifold with bounded mean
curvature, then one can define the stratification S^k(I) = {x:
no tangent cone at x is k+1 symmetric}. It was
proven by Federer that dimS^k<=k, where the dimension is in
the Hausdorff sense. If I^m is further assumed to be a
Z_2-minimizing varifold, then it was proven by Simon that
S^{m-2} is rectifiable. However, besides these results
nothing else about the structure of the singular set is
known. In [NV15_2]
we prove for an integral varifold with bounded mean curvature,
in any manifold, that S^k is k-rectifiable for all k. More
than that, we see that for k a.e. point x\in S^k that there
exists a unique k-plane V such that 'every' tangent cone at x is
of the form VxC for some cone C. That is, a.e. point of
S^k has maximal symmetries, and the k-plane of symmetry is well
defined independent of the tangent cone. If I is further
assumed to be a minimizing hypersurface, we prove that Sing(I)
is rectifiable with effective m-7 Hausdroff measure
bounds. More analytically, we show that the second
fundamental form |A| has apriori estimates in weak L^7.
This is sharp, as the Simons and Lawson cones have the property
that |A| is not even locally in L^7. As in the harmonic
maps case, the techniques of this paper include a new
W^{1,p}-Reifenberg and rectifiable-Reifenberg theorem, which are
of some independent interest. Roughly, these give criteria
which ensure that a set is rectifiable with measure
estimates. In order to apply these theorems to the
singular set of an integral varifold, we prove new L^2-subspace
approximation theorems for integral varifolds with bounded mean
curvature.

Other Applications of Quantitative Stratification and
Rectifiability Methods: The quantitative
stratification was introduced in [ChN] in the context
of Ricci bounds in order to give some effective estimates on
singular sets and solutions. The techniques of [ChN] have since
been extended to harmonic maps and minimal surfaces [ChN2], mean
curvature flows [ChHaNa],
harmonic map flows [ChHaNa2],
critical sets of elliptic equations [ChNaVa], biharmonic
maps [BL],
Q-valued harmonic maps [FMS], etc...
The techniques of were vastly extended in in order
to prove the rectifiable and finiteness estimates. These
techniques have also recently been extended to the case of
stationary Yang Mills in [W].

Regularity and Bounded Ricci Curvature:

Codimension Four Conjecture: Together with Jeff
Cheeger, in [ChN2]
we proved the codimension four conjecture. Roughly, we
show that a metric space X which is a Gromov-Hausdorff limit of
noncollapsed manifolds with bounded Ricci curvature must be
smooth away from a set of codimension four. Combining this
with the ideas of quantitative stratification we prove L^p
estimates for manifolds with bounded Ricci curvature for all
p<2. One of the primary innovations toward the
proof is a new transformation estimate. In short, what
makes the problem difficult are in fact the codimension two
singularities. Since this is a set of capacity zero,
standard estimates for harmonic functions have a hard time
seeing them. To solve this, it is necessary to estimate
our harmonic functions modulo transformation of the image.
This allows us to see into a set whose size is just big enough
to rule out the existence of codimension two
singularities. Ruling out codimension three singularities
is comparatively simple.

Dimension Four and Finite Diffeomorphism Conjecture:
In [ChN2] we
are also able to improve the results in dimension four. We
show the collection of four manifolds with bounded Ricci,
diameter, and volume have at most a finite number of
diffeomorphism classes, solving a conjecture of Anderson.
A local version of this proves L^2 bounds for four manifolds
with bounded Ricci curvature. The proof is in that
order, in that we prove the L^2 curvature bound by first showing
the diffeomorphism finiteness, and then combine this with some
local estimates and a Chern-Gauss-Bonnet formula to conclude the
curvature bound.

L^2 Conjecture and n-4 Finiteness Conjecture: Together
with Wenshuai Jiang, in [JW] we prove that
a noncollapsed manifold with bounded Ricci curvature must have
an apriori L^2 bound on its curvature tensor. As a
preliminary result we show that the singular set of a limit of
such spaces has finite n-4 Hausdorff measure, which is a
strengthening of the codimension four results and proves a
conjecture of Cheeger-Colding. The proof involves several
new techniques, including a neck decomposition theorem which
decomposes the manifold into so called neck regions and
\eps-regularity regions. The power of the decomposition
are the effective n-4 content estimates on the number of pieces,
which is crucial to L^2 question and the H^{n-4} finiteness of
the singular set. The L^2 estimate then follows from being
able to prove the corresponding estimate on each neck region,
which itself requires a new superconvexity estimate, which gives
a growth and decay estimate on the harmonic functions on neck
regions.

eps-Regularity in the Collapsed Case: Together
with Ruobing Zhang, in [NZ] we prove the
first eps-regularity theorems for spaces with collapsed spaces
with bounded Ricci curvature based on Gromov-Hausdorff
behavior. In the noncollapsed case it is known that if a
ball is GH close to a ball in R^n, then that ball must be smooth
close to a ball in R^n. In the collapsed case where the
ball is close to some R^k for k<n, or even R^kxZ for some
metric space Z, this need not be true. However, we prove
that either the ball is smooth, or there is a topological
obstruction in the fundamental group of that ball, giving the
dichotomy that either an eps-regularity holds or there is a
topological obstruction.

Structure and Regularity of Lower Ricci
Curvature:

Constant Dimension and Isometry
Group=Lie Group Conjectures:In
[CN] we
proved the constant dimension conjecture, which roughly
state that limit spaces of manifolds with lower Ricci
curvature bounds have a well defined dimension, and we
proved the conjecture that the isometry groups of such
spaces are lie groups. In particular, you may not
have limits which look like the sum of manifolds of
differing dimensions. The proofs revolve around
new estimates, which in short state that the geometry of
metric balls can only change at a Holder rate along
minimizing geodesics. This Holder rate turns out
to be sharp, and we provide examples to show this.

Examples with Lower Ricci
Curvature: In a different direction it is also
important to understand to what extent examples exist which are
as degenerate as possible. Unlike the Alexandrov case,
e.g. limits with lower sectional curvature, it is not the case
that tangent cones even need to be unique anymore, though it is
always an open question as to what extent this nonuniqueness can
be pushed. In [CN2]
we give a characterization for the families of tangent cones
which may appear at a point in a noncollapsed limit. We
have two primary applications of this. First we construct
limit spaces whose tangent cones at a point have singular sets
of varying dimensions. In particular, this rules out the
possibility of stratifying limit spaces based on tangent cones.
Secondly we construct examples where differing
tangent cones at a point may not even be homeomorphic. In
[CN3] we provide
further examples of degenerate limit spaces, as well as prove a
Lipschitz structure theorem for Reifenberg spaces.

Characterizing Bounded Ricci Curvature and Path
Space:

Smooth Spaces with Bounded Ricci Curvature: In [N13] new estimates
are proved for spaces with bounded Ricci curvature, and these
estimates are proved to be equivalent to bounded Ricci
curvature. The first such method is an infinite
dimensional generalization of the Bakry-Emery gradient estimates
on path space P(M) of the manifold M. The second method
studies the 1-parameter decomposition of functions on path space
P(M) determined by martingales, and in particular shows that
bounded Ricci curvature is equivalent to certain C^{1/2}-Holder
estimates. The third method studies a family of
Ornstein-Uhlenbeck operators on path space, a form of infinite
dimensional laplacian, and proves that bounds on the Ricci
curvature are equivalent to a spectral gap on these operators.

NonSmooth Spaces with Bounded Ricci Curvature:
Using the above as motivation, in [N13] we define the
notion of bounded Ricci curvature for a general metric-measure
space X. We prove a variety of structure theorems about
such spaces, in particular that one can still do analysis on the
path space P(X) of such spaces. We show such spaces have a
lower Ricci curvature bound in the sense of Lott-Villani-Sturm.

Bochner Formula for Martingales: In [NH16] Bob
Haslhofer and I construct a new type of Bochner formula for
martingales on the path space of a manifold. This Bochner
formula turns out to be related to two-sided bounds on Ricci
curvature in much the same way the standard Bochner formula is
related to lower bounds on Ricci curvature. The method not
only gives rise to (much) simpler and more natural proofs of the
estimates of [N13],
but may be used to provide a variety of strengthened estimates,
including new L^2 hessian estimates on martingales, which seems
to us to be new even in R^n.

Characterizations of Ricci Flow: In [HN15] we prove
new estimates on Ricci flows. These estimates generalize
the ideas of [N13]
from the elliptic context of Einstein manifolds to the parabolic
context of the Ricci flow, by studying estimates on the
spacetime path space of a Ricci flow. The estimates are
not only new for a Ricci flow, but it is shown that if a one
parameter family of Riemannian manifolds satisfy these
estimates, then the family must solve the Ricci flow.
Thus, these estimates are sharp enough to characterize the Ricci
flow.

Critical Sets of Elliptic Equations:
Joint with Daniele Valtorta we study critical sets of elliptic
equations on manifolds. If the coefficients are smooth it
has been standard that the critical set has apriori n-2
Hausdorff measure estimates, however even under only C^k bounds
on the coefficients with k<\infty this has been an open
question. In [NaVa]
we develop new techniques for estimating such sets, and prove
the desired n-2 Hausdroff measure estimates on the critical sets
of elliptic equations under only Lipschitz coefficients.
The regularity assumption on the coefficients is sharp.
Indeed, our estimates are much stronger and actually prove
estimates for the volumes of tubes around the effective critical
set.

Ricci Solitons and
Ricci Flow: Ricci solitons can be
viewed as a generalization of Einstein manifolds. My
primary results in this area are for shrinking solitons (the
ricci soliton equivalent of having positive einstein constant).
In [N] a
classification is given for four dimensional shrinking solitons
with bounded nonnegative curvature. In this process it is
shown that shrinking solitons with bounded curvature are apriori
noncollapsed and gradient. For the Ricci flow, together
with Hans Hein, we proved in [HN] universal
log-Sobolev estimates for the conjugate heat kernel along
the Ricci flow. We used this to prove new
epsilon-regularity results for the Ricci flow.

Sectional Curvature:
Joint with Gang Tian we prove two collections of results
in [NaT1], [NaT2]. First
we analyze the orbifolds behavior of limits of n-manifolds with
bounded sectional curvature, and prove that such limit spaces
are Riemannian orbifolds away from a set of Hausdorff dimension
n-5. In particular, limits of four manifolds with
bounded sectional curvature are always Riemannian orbifolds.
We apply this to study limits which only have bounded
Ricci curvature. Secondly, we construct a new structure on
limit spaces with bounded sectional, which we call an
N^*-bundle. This structure, as opposed to the N-structure,
lives on the limit space itself and can in many ways be viewed
as a dual structure whose purpose is primarily for doing
analysis. As applications we prove generalizations
of finite diffeomorphism theorems due to Anderson, as well as a
stability theorem for pinched Ricci curvature. Namely, it
turns out that for a collapsing sequence with bounded diameter
and sectional curvature if the Ricci curvature is tending
to zero, then the limit space is a Ricci flat orbifold.
Notice that in general limits with bounded sectional
can have much worse than orbifold behavior, and conversely Ricci
curvature bounds not be preserved when bubbling occurs.
The proof relies on a maximum principle with the
N^*-bundle. The maximum principle nature of the proof is
quite necessary, as if even the diameter assumption is dropped
then the result is not correct even if you assume the full
sectional curvature is tending to zero.